I recently had a conversation with one mathematician who reckons that all sorts of combinatorial results are nothing compared to the things done in the algebraic geometry. As I do not have any expertise in AG, I have the following question:

Is AG really that important?

Surfing over the book of Hartshorne, one may get a feel that this is just some bunch of definitions with the hope to be able to do some big problems. First steps of AG, as far as I have been told, indeed lead to solutions, but is AG still natural, or just a trend in Maths (for producing Fields medalists).

To get things clear, I have nothing against AG, but it really got me upset that some mathematicians do not count such things as results about the random graph as Maths.

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Condescending people are annoying everywhere. In mathematics people would often see their field as "THE" tool for doing mathematics, but this is really just "Everything looks like a nail when you're hold a hammer." variation. I do agree, however, that certain problems in mathematics can be easily solved with different tools.
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Asaf KaragilaJan 29 '12 at 19:14

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Victor, I voted to close as "subjective and argumentative", because discussions of this question are likely to be both of those things, hence off-topic for MO.
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Tim PerutzJan 29 '12 at 19:22

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I think this question would have more potential if it was worded differently. Rather then asking if algebraic geometry is "just a trend in maths (for producing Fields medalists)" the following two questions may be likely to elicit deeper responses: "What motivated the study of Algebraic Geometry?" and "What problems in other areas of mathematics have been solved by using Algebraic Geometry?"
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Eric NaslundJan 29 '12 at 19:24

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It's one instance of the rough subdivision of mathematicians between "theory builders" and "problem solvers": both are doing natural and important mathematics, but sometimes they can't fully appreciate each other's viewpoint.
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QfwfqJan 29 '12 at 19:40

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(btw, voted to close as subjective and argumentative)
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QfwfqJan 29 '12 at 19:41

2 Answers
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Algebraic geometry is the study of spaces defined by polynomial equations. So on the one hand it provides examples of interest for geometers and topologists, and on the other hand it lets algebraists and number theorists take a geometric perspective on their subjects.

Algebraic geometry has been an important part of mathematics since Descartes, who pretty much invented it. In other words it is part of 17th century mathematics, like calculus. It happens that there is more left to do in algebraic geometry than in calculus (in terms Newton, who worked on cubic curves, would understand); and that some problems that are apparently in calculus, for example indefinite integration by closed formulae, are really better understood geometrically, as became apparent in the 19th century. We are still working out where the ideas of Hilbert and Poincaré leave us in expressing methods and solutions in algebraic geometry, in fact. A very active 20th century didn't settle that.